Lyapunov-Schmidt Reduction for Unfolding Heteroclinic Networks of Equilibria and Periodic Orbits with Tangencies

TL;DR

This paper develops a Lyapunov-Schmidt reduction method for analyzing complex heteroclinic networks involving equilibria and periodic orbits, including tangencies, providing algebraic conditions for solutions near the network.

Contribution

It introduces a novel reduction technique that handles heteroclinic networks with tangencies and mixed vertices, extending previous methods to more general and intricate systems.

Findings

Reduction to algebraic equations for solutions near heteroclinic networks

Conjugacy to shift dynamics for homoclinic orbits

Analysis of equilibrium-to-periodic heteroclinic cycles

Abstract

This article concerns arbitrary finite heteroclinic networks in any phase space dimension whose vertices can be a random mixture of equilibria and periodic orbits. In addition, tangencies in the intersection of un/stable manifolds are allowed. The main result is a reduction to algebraic equations of the problem to find all solutions that are close to the heteroclinic network for all time, and their parameter values. A leading order expansion is given in terms of the time spent near vertices and, if applicable, the location on the non-trivial tangent directions. The only difference between a periodic orbit and an equilibrium is that the time parameter is discrete for a periodic orbit. The essential assumptions are hyperbolicity of the vertices and transversality of parameters. Using the result, conjugacy to shift dynamics for a generic homoclinic orbit to a periodic orbit is proven. Finally, equilibrium-to-periodic orbit heteroclinic cycles of various types are considered.